13.4 - Cross Product

You can express vectors as matrices

<1,3,6>
<2,1,4>     Is the same as  --------------------->
<1,1,1>


Determinant: A function that assigns a numerical value to a square array of numbers. (2x2, 3x3, etc)

In 2-space the determinant is the area of the parallelogram formed by two vectors.

In 3-space the determinant is related to the volume of the parallel pipet formed by three vectors.

How to calculate the determinant

---

Step 1:

Take the top of the left column, cross out the rest of the row and column

Then take the 2x2 vector remaining, multiply the upper left with the lower right, and subtract the upper right multiplied by the lower left. Multiply the result by the circled number. In this case you would do:

((4*2)-(-4*3))(3)

Step 2:

Repeat the same step, using the center of the top row:

((1*2)-(-4*0))(-2)

Step 3:

Repeat the same step, now using the upper right corner:

((1*3)-(4*0))(-5)

Finally, add the results from the last three steps.

Note, alternating steps are subtracted, resulting in:

(3)(8 + 12) - (-2)(2 - 0) + (-5)(3 - 0) = 49

Note:

• Lines are used to denote determinants, while [brackets] denote matrices.
• If two rows of a matrix are the same the determinant is zero.
• If two rows of a matrix are switched the determinant is multiplied by -1.

To check your work, you can also enter the matrices into the calculator in:
2nd -> MATRX -> EDIT
Once entered, run 2nd -> MATRX -> MATH -> det( [A] ) to calculate the determinant!


Cross Products

---

If U = <u1, u2, u3> and V = <v1, v2, v3> then the cross product of U and V (UxV) is defined as:

Properties of determinants and cross products

---

The cross product of two vectors is a third vector in a perpendicular direction, and it has a length equal to that of the parallelogram formed by the two original vectors.

Example:

Find the area of the parallelogram formed by A = <1,2,3> and B = <-1, 1, -2>

A x B = <-7, -1, -3>

|| <-7, -1, -3> || = √( 59 )

To find the area of the triangle between the vectors, just cut the parallelogram in half!

Note:  You must take the absolute value of the vector length, because areas and volumes are always positive.

Scalar Triple Product

---

The same concept can be applied in 3-Space, when you have a parallel pipet formed by three vectors.

Given vectors A, B, and C, the volume of the pipet is: (A x B) • C

Example: Use a scalar triple product to show that the vectors A <1,4,-7> B <2,-1,4> and C <0,-9,18> are coplanar.

Note: If three vectors are coplanar then the volume will be zero, because the shape will be flat.

<18, -36, -18> • <1, 4, -7> = 0

Relations with cross and dot products:

U • V = ||U|| ||V|| cos ø

|| U x V || = ||U|| ||V|| sin ø

Calculator Note:
Below is the source code to a calculator program which does cross products for you! Use it for good:

Disp "VCTR A"
Input A
Input B
Input C
Disp "VCTR B"
Input D
Input E
Input F
Disp ((B*F)-(C*E))
Disp (-1)((A*F)-(C*D))
Disp (A*E)-(B*D)

XOXO, 
Milo and Allison